The first step is to solve $\displaystyle y'''+6y''+12y'+8y = 0$ the charachteristic equation is $\displaystyle k^3 + 6k^2 + 12k+8 = 0$. This factors as $\displaystyle (k+2)^{2} = 0$. Thus, the general solution is $\displaystyle c_1e^{-2x}+c_2xe^{-2x}+c_3x^2e^{-2x}$. To find the particular solution we would look for a solution of the form $\displaystyle Ae^{-2x}$ but since this is included amongst the general we need to look for a solution of the form $\displaystyle Ax^3e^{-2x}$.